THE NUCLEUS FOR RESTRICTED LIE ALGEBRAS

DAVID J. BENSON AND DANIEL K. NAKANO

1 Introduction

1.1 The concept of the nucleus was first introduced in [BCR] in the context of the representation theory and cohomology of a finite group G over a field k of characteristic p dividing |G|. The authors were attempting to prove an old conjecture, which states that if S is a simple kG-module in the principal block then H^{•}(G, S) ≠ 0. This conjecture remains open, but the nucleus has become an important part of the machinery for understanding the relationship between cohomology and representation theory. For example, it is an essential ingredient in the investigation of the thick subcategories of the stable module category [BCRi3, C2]. The nucleus also makes an appearance when we try to understand the annihilator of Ext_{kG}^{•}(M, M′) in terms of the varieties of M and M′ [C1].

There are two different versions of the nucleus in the representation theory of finite groups. When the word “nucleus” is used without qualification, it refers to the group theoretical nucleus Y_{G}. The other version is called the representation theoretical nucleus θ_{G}. They are both defined as subsets of the cohomology variety V_{G}, and one of the main theorems of the subject [Ben2] states that these subsets are equal: Y_{G} = θ_{G}. This theorem indicates further how the group structure is related directly to the representation theory and cohomology. It is easy to see from the definition that the subset Y_{G} is a closed homogeneous subvariety of V_{G}. The same is not at all true of θ_{G}, but it may be seen as a consequence of the above theorem.

To be more explicit, the cohomology variety V_{G} is defined to be the maximal ideal spectrum of the cohomology ring H^{•}(G, k) if p = 2, and of the even degree part of the cohomology ring if p > 2. The structure of this variety was completely described by Quillen [Qu] in terms of the set of elementary abelian p-subgroups E of G and the conjugations and inclusions between them. If M is a finitely generated kG-module then V_{G}(M) is defined to be the closed homogeneous subvariety of V_{G} determined by the kernel of the map

H^{•}(G, k) = Ext_{kG}^{•}(k, k) → Ext_{kG}^{•}(M, M)

given by tensoring exact sequences over k with M.

The nucleus Y_{G} is defined to be the union of the images of the maps res_{G,E}^{∗}: V_{E} → V_{G}, where E runs over the set of elementary abelian p-subgroups of G with the property that the centralizer C_{G}(E) is not p-nilpotent. The representation theoretic nucleus θ_{G} is defined to be the union of the subvarieties V_{G}(M) as M runs over the finitely generated modules in the principal block such that H^{•}(G, M) = 0 for • > 0. A description which generalizes to other blocks is that θ_{G} is equal to the union of the subvarieties V_{G}(M)∩V_{G}(N) as M and N run over the finitely generated modules in the principal block such that Ext_{kG}^{•}(M, N) = 0 for • > 0 (see [Ben3]).

Date: April 2001.

Research of the first author partially supported by NSF grant DMS-9988110.

Research of the second author partially supported by NSF grant DMS-9800960.

1.2 Our main goal in this paper will be to investigate the corresponding definitions for the p-restricted Lie algebras. In this case, the cohomological variety V_{𝔤} of 𝔤 was described by Suslin, Friedlander and Bendel [SFB1]. It is homeomorphic to the subvariety of the vector space 𝔤 given by the kernel of the p-restriction map x ↦ x^{[p]}. If G is a connected reductive algebraic group and 𝔤 = Lie G with p > h, the Coxeter number, then V_{𝔤} is the same as the variety N of nilpotent elements in 𝔤.

We begin by computing the representation theoretic nucleus in the case of a p-nilpotent Lie algebra, where the answer is {0}. This is analogous to the case of a finite p-group, where the trivial module is the only simple module. For a Borel subalgebra 𝔟 of 𝔤, the nucleus is the entire cohomological variety. This is somewhat surprising because for finite groups, in odd characteristic the nucleus is always a proper subvariety of the cohomology variety [CR]. This example provides strong indications that there is are significant qualitative differences between the group case and the Lie algebra case. Furthermore, in this case there is only one block, and the one dimensional simple module which induces to the Steinberg module is an example of a simple module with no cohomology. So the original motivating question for finite groups has a negative answer in the case of p-restricted Lie algebras. In [CNP, Thm 4.3], it was shown that there exists non-projective indecomposable modules in the principal block with no cohomology as long as 𝔤 does not have underlying root system A_{1}. Our main theorem, which we prove in Section 3.5, extends this result. We provide an identification of the representation theoretic nucleus for the restricted Lie algebra of a connected reductive algebraic group, under the hypothesis that p > h.

(1.2.1) Theorem. Let G be a connected reductive algebraic group over a field of characteristic p > h, with p-restricted Lie algebra 𝔤. Then the representation theoretic nucleus θ_{𝔤} is equal to the subregular orbit.

The subregular orbit is the (unique) largest closed orbit of G on N which is not equal to the whole of N. This theorem provides evidence that for an arbitrary restricted Lie algebra 𝔤 there may be closed subvariety Y_{𝔤} related to the structure of the Lie algebra which equal θ_{𝔤}. The question of the correct definition for Y_{𝔤} in general for p-restricted Lie algebras 𝔤 remains open. The proof of the above theorem uses in an essential way the recent calculation of the varieties of Weyl modules given in [NPV] as well as the prominent translation functors due to Jantzen.

We end with some questions suggested by our investigations of the representation theoretic nucleus.

2 Preliminaries

2.1 Notation: The conventions throughout this paper will closely follow those in [Jan]. Let G be a reductive connected algebraic group defined and split over the finite field 𝔽_{p} with p elements. Moreover, let k be the algebraic closure of 𝔽_{p}. We will often consider G as an algebraic group scheme over k. Let T be a maximal split torus and Φ be a root system associated to G for a Euclidean space 𝔼. The inner product on 𝔼 will be denoted by ⟨ , ⟩ and the positive (resp. negative) roots will be denoted by Φ^{+} (resp. Φ^{−}). We shall fix a base of simple roots Δ. Let α^{∨} = 2α/⟨α, α⟩ be the coroot corresponding to α ∈ Φ and α_{0} be the highest short root. Set ρ equal to the half sum of positive roots. For any α ∈ Φ, set d_{α} = ⟨α, α⟩/⟨α_{0}, α_{0}⟩. The Coxeter number associated to Φ is h = ⟨ρ, α_{0}^{∨}⟩ + 1. Let W be the Weyl group corresponding to Φ and W_{p} be the affine Weyl group. Let B be a Borel subgroup containing T corresponding to the positive roots and U be the unipotent radical of B.

For an affine group scheme H over k, let F : H → H^{(1)} be the Frobenius morphism and F^{r} be the composition of the Frobenius morphism r times with itself. Set H_{r} = ker F^{r}. For r = 1, it is well-known that there exists a categorical equivalence between H_{1}-modules and restricted Lie(H)-modules (or equivalently modules for the restricted enveloping algebra u(Lie(H)). This fact will be used extensively throughout this paper. In particular, we will be interested in the cases when H = G, B, and U. For T a maximal split torus in G, set G_{r}T = (F^{r})^{−1}(T).

Let X(T) be the integral weight lattice obtained from Φ. The set X(T) has a partial ordering defined as follows: λ ≥ μ if and only if λ − μ ∈ ∑_{α∈Δ} ℕα. The set of dominant integral weights is defined by

X(T)_{+} = {λ ∈ X(T) : 0 ≤ ⟨λ, α^{∨}⟩ for all α ∈ Δ}.

For λ ∈ X(T)_{+}, let H^{0}(λ) = ind_{B}^{G}λ and L(λ) = soc_{G}H^{0}(λ). This provides a one-to-one correspondence between finite-dimensional simple G-modules (labelled by L(λ)) with X(T)_{+}. Furthermore, the set of restricted weights is

X_{r}(T) = {λ ∈ X(T) : 0 ≤ ⟨λ, α^{∨}⟩ < p^{r} for all α ∈ Δ}.

For λ ∈ X_{r}(T) let L_{r}(λ) = L(λ). Furthermore, if λ ∈ X(T) we can express λ = λ_{0} + p^{r}λ′ where λ_{0} ∈ X_{r}(T) and λ′ ∈ X(T). Set L̂_{r}(λ) = L(λ_{0}) ⊗ p^{r}λ′. A complete set of representatives for the isomorphism classes of simple G_{r}-modules (resp. G_{r}T -modules) is given by {L_{r}(λ): λ ∈ X_{r}(T)} (resp. {L̂_{r}(λ): λ ∈ X(T)}). The Weyl modules are given by V (λ) = H^{0}(−w_{0}λ)^{∗} for λ ∈ X(T)_{+}. Here w_{0} is the long element in W and ^{∗} represents taking the dual module.

2.2 Support varieties: Let A be a finite k-group and let

H(A, k) = \begin{cases}H^{2•}(A, k) & if char k ≠ 2 \\ H^{•}(A, k) & if char k = 2. \end{cases}

According to [FS], the cohomology ring R = H(A, k) is a commutative, finitely generated k-algebra. For finite dimensional M, M′ ∈ A-mod, define the relative support variety V_{A}(M, M′) as follows. Yoneda composition defines an action of R on Ext_{A}^{•}(M, M′). Let J = J_{A}(M, M′) be the annihilator ideal in H(A, k) for this action. Set V_{K}(M, M′) equal to the maximum ideal spectrum of R/J. The support variety V_{A}(M) is defined by setting V_{A}(M) = V_{A}(M, M).

Let A_{0} be the block of A containing the trivial module. Set

S = {M ∈ mod(A_{0}) : H^{•}(A, M) = 0, • > 0}
= {M ∈ mod(A_{0}) : V_{A}(k, M) = {0}}.

The representation theoretic nucleus of A, θ_{A}, is defined as follows:

θ_{A} = ⋃_{M∈S} V_{A}(M).

From the definition, it is not clear that θ_{A} is a closed set (i.e., a variety). For finite groups, the representation theoretic nucleus is a variety because one can identify this set with the union Y_{G} of the images of restriction maps from elementary abelian p-subgroups whose centralizers are not p-nilpotent [Ben2, Cor. 1.3]. In our computations involving infinitesimal Frobenius kernels we will show that the representation theoretic nucleus in these cases is also a variety.

Let H be an algebraic group scheme over k and 𝔥 = Lie H. The Lie algebra 𝔥 is a restricted Lie algebra with p-mapping x → x^{[p]}. According to [SFB1, (1.6), (5.11)], V_{H_{1}}(k) is homeomorphic to N (𝔥) := {x ∈ 𝔥 : x^{[p]} = 0 }. Furthermore, under this identification, if M is an H_{1}-module, then V_{H_{1}}(M) is homeomorphic to

{x ∈ 𝔥 : x^{[p]} = 0, M_{<x>} is not free}^{(1)} ∪ {0}.

We remark that if M is an H-module, then V_{H_{1}}(M) is an H-stable variety of 𝔥 under conjugation.

2.3 Orbit theory: Let G be a reductive algebraic group with 𝔤 = Lie G. Let N be the variety of nilpotent elements in 𝔤. For p > h, N = N (𝔤). The group G acts on N by conjugation and N (𝔤) is a G-stable subvariety of N.

The orbit theory of G on N has been well-studied. We refer the reader to [CM] [Hum] for details. The variety N has finitely many G-orbits. These orbits have been classified. For each α ∈ Φ, let x_{α} be a root vector. Set x_{reg} = ∑_{α∈Δ} x_{α}. Then

N = G · 𝔲 = \overline{G · x_{reg}}

is called the regular orbit.

Let I ⊆ Δ. Then 𝔤 = 𝔲_{I}^{+} ⊕ 𝔩_{I} ⊕ 𝔲_{I} has a Levi decomposition. The set G · 𝔲_{I} can be expressed as the closure of an orbit. Orbits of this form are called Richardson orbits. We should remark that Levi factors which are conjugate under the action of G give rise to the same Richardson orbit. In a specific case one can take J = {α} ⊆ Δ. Then 𝔩_{J} is of type A_{1}. The Richardson orbit in this special case is called the subregular orbit. The subregular orbit is the largest proper orbit (i.e. every proper orbit is contained in the subregular orbit). For I ⊆ Δ, let Φ_{I} be the root system in Φ generated by I. Moreover, for λ ∈ X(T), let

Φ_{λ,p} = {α ∈ Φ : d_{α}⟨λ + ρ, α^{∨}⟩ ∈ pℤ}.

The following result [NPV, (6.2.1) Thm.] due to Parshall, Vella and the second author involving Richardson orbits will be essential in our computations of the representation theoretic nucleus.

(2.3.1) Theorem. For λ ∈ X(T)_{+} and p good, choose I ⊆ Δ so that w(Φ_{λ,p}) = Φ_{I} for some w ∈ W. Then V_{G_{1}}(H^{0}(λ)) ⊆ G · 𝔲_{I}.

2.4 Translation functors: Let \overline{C}_{ℤ} = \{λ ∈ X(T) : 0 ≤ ⟨λ + ρ, α^{∨}⟩ for all α ∈ Φ\}. Moreover, let M be a G-module and let pr_{σ}(M) be the sum of all submodules of M with G-composition factors with high weight in W_{p} · σ. If Z is a set of W_{p}-orbits in X(T) (under the dot action) then

M ≅ ⨁_{μ∈Z} pr_{μ}(M).  (2.4.1)

We will use the translation functors [Jan, II 7.6] first constructed by Jantzen. If λ, μ ∈ \overline{C}_{ℤ}, let ν_{1} ∈ W (μ − λ) ∩ X(T)_{+}. For any G-module M, let

T_{λ}^{μ}(M) = pr_{μ}(pr_{λ}(M) ⊗ H^{0}(ν_{1})).

The functor T_{λ}^{μ} is exact. Moreover, T_{λ}^{μ} and T_{μ}^{λ} are adjoint to each other. The functor T_{λ}^{μ} commutes with the forgetful functor from G-modules to G_{r}T -modules. Consequently, T_{λ}^{μ} and T_{μ}^{λ} are adjoint in the category of G_{r}T -modules. Many of the results for G-modules (in [Jan, II Ch. 7]) have analogous statements in the category of G_{r}T -modules (see [Jan, II 9.19]). We will rely on these results in our computations. Finally observe that since V_{G_{1}}(M_{1} ⊗ M_{2}) = V_{G_{1}}(M_{1}) ∩ V_{G_{1}}(M_{2}) and V_{G_{1}}(M_{1} ⊕ M_{2}) = V_{G_{1}}(M_{1}) ∪ V_{G_{1}}(M_{2}), we have

V_{G_{1}}(T_{λ}^{μ}(M)) ⊆ V_{G_{1}}(M) ∩ V_{G_{1}}(H^{0}(ν_{1})) ⊆ V_{G_{1}}(M).

3 Computation of the representation theoretic nucleus

3.1 Carlson and Robinson [CR, Thm. 4.1] have demonstrated that for group algebras of a finite group, the nucleus must be a proper subvariety of support variety of the trivial module if the underlying field does not have characteristic two. For fields of characterstic two there are examples where the nucleus is not proper [CR, Prop. 5.2]. In our first result we show that for the Borel subalgebra of a classical Lie algebra the representation theoretic nucleus need not be proper. This result is independent of the characterstic of the field.

(3.1.1) Proposition. Let G be a connected reductive algebraic group. Then

(a) θ_{U_{1}} = {0};

(b) θ_{B_{1}} = N (𝔲)

In particular if p > h then θ_{B_{1}} = 𝔲.

Proof. (a) Since 𝔲 is a p-nilpotent Lie algebra, the only simple U_{1}-module is the trivial module k. If H^{•}(U_{1}, M) = 0 for • > 0 then M is a projective U_{1}-module, thus θ_{U_{1}} = {0}.

(b) Let (p−1)ρ ∈ X(T)_{+} be the dominant weight corresponding to the Steinberg module. We have

L((p − 1)ρ) = ind_{B_{1}}^{G_{1}}(p − 1)ρ.

Since L((p − 1)ρ) is a projective G_{1}-module, it follows that

H^{•}(B_{1}, (p − 1)ρ) = H^{•}(G_{1}, L((p − 1)ρ)) = 0

for • > 0. The group scheme B_{1} has one block. Moreover, (p − 1)ρ is a one-dimensional B_{1}-module so V_{B_{1}}((p − 1)ρ) = N (𝔟) = N (𝔲). Consequently, θ_{B_{1}} = N (𝔲). _{□}

3.2 Let ↑ be the Strong Linkage relation on X(T) as defined in [Jan, II 6.4]: λ ↑ μ if and only if there exists μ_{1}, μ_{2}, …, μ_{t} ∈ X(T) and reflections s_{1}, s_{2}, …, s_{t+1} ∈ W_{p} such that

λ ≤ s_{1} · λ = μ_{1} ≤ s_{2} · μ_{1} = μ_{2} ≤ ⋯ ≤ s_{t} · μ_{t−1} = μ_{t} ≤ s_{t+1} · μ_{t} = μ.

Here · denotes the “dot action” of the affine Weyl group.

Let B_{λ}(G_{1}) (resp. B_{λ}(G_{1}T)) be the block for G_{1} (resp. G_{1}T) that contains L_{1}(λ) (resp. L̂_{1}(λ)). For brevity we will write μ ∈ B_{λ}(G_{1}) to indicate that the simple G_{1}-module L_{1}(μ) is in the block B_{λ}(G_{1}). A similar statement can be made for λ ∈ B_{0}(G_{1}T). The principal block for G_{1} (resp. G_{1}T) is B_{0}(G_{1}) (resp. B_{0}(G_{1}T)). From [Jan, II 9.19 (1)], λ ∈ B_{0}(G_{1}T) if and only if λ ∈ W ·0+pX(T). Moreover, λ ∈ B_{0}(G_{1}) (λ ∈ X_{1}(T)) if and only if λ ≡ W ·0 (mod pX(T)).

(3.2.1) Proposition. If λ ∈ B_{0}(G_{1}) (λ ∈ X_{1}(T)) then there exists ν ∈ X(T) such that

(i) 0 ↑ λ + pν; (ii) λ + pν ∈ X(T)_{+} with ν ∈ X(T)_{+}.

Proof. Since λ ∈ B_{0}(G_{1}), there exists w ∈ W, ζ ∈ X(T) such that w · λ + pζ = 0, so w · λ = −pζ. From [Jan, II 6.4(5)], we have w · λ ↑ λ for all w ∈ W. This implies that −pζ ↑ λ or 0 ↑ λ + pζ. Let β be the highest (long) root. By applying [Jan, II 6.4(4)], λ + pζ ↑ λ + pζ + psβ for s ∈ ℤ^{+}. Therefore, 0 ↑ λ + pζ + psβ for s ∈ ℤ^{+}. For s sufficiently large ζ + sβ ∈ X(T)_{+} with λ + p(ζ + sβ) ∈ X(T)_{+} because ⟨β, α^{∨}⟩ > 0 for all α ∈ Δ. _{□}

3.3 The following theorem shows that for classical Lie algebras for p ≥ h, the representation theoretic nucleus is contained in the subregular nilpotent orbit, and is thus a proper subvariety of the support variety of the trivial module.

(3.3.1) Theorem. Let G be a connected reductive algebraic group with p ≥ h. Then θ_{G_{1}} ⊆ G · 𝔲_{J} where J = {α} ⊆ Δ.

Proof. Let D = {σ ∈ X(T)_{+} : 0 ↑ σ} and S = {M ∈ mod(B_{0}(G_{1})) : H^{•}(G_{1}, M) = 0, • > 0}. We will first show that V_{G_{1}}(H^{0}(σ), M) ⊆ G · 𝔲_{J} for all σ ∈ D and M ∈ S. This will be accomplished by using induction on the ordering in D. First observe that Ext_{G_{1}}^{•}(H^{0}(0), M) = Ext_{G_{1}}^{•}(k, M) = 0 for • > 0, thus V_{G_{1}}(H^{0}(0), M) = {0} ⊆ G · 𝔲_{J}.

Now let σ ∈ D, σ ≠ 0 and assume that for all η ∈ D with 0 ↑ η and η < σ, V_{G_{1}}(H^{0}(η), M) ⊆ G · 𝔲_{J}. Since σ ≠ 0 and 0 ↑ σ, there exists a reflection s in W_{p} such that s · σ < σ and 0 ↑ s · σ. Let μ′ be such that Stab_{W_{p}}μ′ = {1, s}. There exists a w ∈ W_{p} such that for some λ, μ ∈ \overline{C}_{ℤ}, w · λ = σ and w · μ = μ′. According to [Jan, II 7.13 Prop.], the module T_{μ}^{λ}(H^{0}(μ′)) = T_{μ}^{λ}(H^{0}(w · μ)) has composition factors with multiplicity at most one of the form H^{0}(σ) and H^{0}(s · σ).

Without loss of generality we may assume that there exists an exact sequence of the form

0 → H^{0}(s · σ) → T_{μ}^{λ}(H^{0}(μ′)) → H^{0}(σ) → 0.

We remark that the order of the composition factors may be reversed, but the argument is the same in either case. Also, note that is possible to have s · σ ∉ X(T)_{+} in which case H^{0}(s·σ) = 0. The short exact sequence above induces a long exact sequence in cohomology:

⋯ → Ext_{G_{1}}^{i}(H^{0}(s · σ), M) → Ext_{G_{1}}^{i}(T_{μ}^{λ}(H^{0}(μ′)), M) → Ext_{G_{1}}^{i}(H^{0}(σ)), M) → …

This implies that

V_{G_{1}}(H^{0}(σ), M) ⊆ V_{G_{1}}(H^{0}(s · σ), M) ∪ V_{G_{1}}(T_{μ}^{λ}(H^{0}(μ′)), M)
⊆ G · 𝔲_{J} ∪ V_{G_{1}}(T_{μ}^{λ}(H^{0}(μ′)), M).

By using the definition of the translation functor and Theorem 2.3.1 with Φ_{μ′,p} of type A_{1}, we have

V_{G_{1}}(T_{μ}^{λ}(H^{0}(μ′)), M) ⊆ V_{G_{1}}(T_{μ}^{λ}(H^{0}(μ′)))
⊆ V_{G_{1}}(H^{0}(μ′))
⊆ G · 𝔲_{J}.

Hence, V_{G_{1}}(H^{0}(σ), M) ⊆ G · 𝔲_{J}.

Next we will show that V_{G_{1}}(L(λ), M) ⊆ G · 𝔲_{J} for all λ ∈ D and M ∈ S. Again we will use induction on the ordering of weights in D. For M ∈ S, one has

V_{G_{1}}(k, M) = V_{G_{1}}(L(0), M) = {0} ⊆ G · 𝔲_{J}.

If λ ∈ D, there exists a short exact sequence of G-modules of the form

0 → L(w · 0) → H^{0}(w · 0) → N → 0

with all composition factors in N of the form L(σ) with σ ↑ λ and σ < λ. By the same argument as in the preceding paragraph (using the long exact sequence), we have

V_{G_{1}}(L(w · 0), M) ⊆ V_{G_{1}}(H^{0}(w · 0), M) ∪ V_{G_{1}}(N, M)
⊆ G · 𝔲_{J} ∪ G · 𝔲_{J}
⊆ G · 𝔲_{J}.

The second to last line follows by using the induction hypothesis and the assertion proved in the preceding paragraph.

Finally, let λ ∈ X_{1}(T) and L_{1}(λ) be a simple G_{1}-module in B_{0}(G_{1}). We claim that V_{G_{1}}(L_{1}(λ), M) ⊆ G · 𝔲_{J} for all M ∈ S. Since λ ∈ B_{0}(G_{1}), by Proposition 3.2.1, there exists ν ∈ X(T)_{+} such that 0 ↑ λ + pν and λ + pν ∈ X(T)_{+}. Therefore,

L(λ + pν) ≅ L(λ) ⊗ L(ν)^{(1)} ≅ L_{1}(λ) ⊗ L(ν)^{(1)}.

This implies that Ext_{G_{1}}^{•}(L(λ + pν), M) ≅ Ext_{G_{1}}^{•}(L_{1}(λ), M) ⊗ L(ν)^{(1)}. Therefore, from the claim in the preceding paragraph,

V_{G_{1}}(L_{1}(λ), M) = V_{G_{1}}(L(λ + pν), M) ⊆ G · 𝔲_{J}

for all M ∈ S. Now suppose that M ∈ S. Then

V_{G_{1}}(M) = V_{G}(⨁_{λ∈B_{0}(G_{1})} L_{1}(λ), M) ⊆ G · 𝔲_{J}.

Consequently, θ_{G_{1}} ⊆ G · 𝔲_{J}. _{□} 3.4 In this section we will investigate the properties of support varieties of induced modules relative to the translation functor.

(3.4.1) Proposition. Let G be a connected reductive algebraic group with p good. Let λ, μ ∈ \overline{C}_{ℤ} such that μ belongs to the closure of the facet containing λ. If w · μ ∈ X(T)_{+} where w ∈ W_{p} then V_{G_{1}}(T_{μ}^{0}(H^{0}(w · μ))) = V_{G_{1}}(H^{0}(w · μ)).

Proof. Set μ′ = w · μ. Since T_{μ}^{λ}(H^{0}(μ′)) is a G_{1}T -module, it follows by [Jan, I 6.9 (4)(5)] for σ ∈ X_{1}(T),

Ext_{G_{1}}^{•}(L_{1}(σ), T_{μ}^{λ}(H^{0}(μ′))) ≅ ⨁_{ν∈X(T)} Ext_{G_{1}T}^{•} (L̂_{1}(σ + pν), T_{μ}^{λ}(H^{0}(μ′)))

≅ ⨁_{ν∈X(T)} Ext_{G_{1}T}^{•} (T_{λ}^{μ}(L̂_{1}(σ + pν)), H^{0}(μ′))

Therefore,

Ext_{G_{1}}^{•}(_{⨁σ∈B_{λ}(G_{1})}L(σ), T_{μ}^{λ}(H^{0}(μ′))) ≅ ⨁_{σ∈B_{λ}(G_{1})} ⨁_{ν∈X(T)} Ext_{G_{1}T}^{•} (T_{λ}^{μ}(L̂_{1}(σ + pν)), H^{0}(μ′)). (3.4.2)

Let γ ∈ X(T) be such that L̂_{1}(γ) ∈ B̂_{μ}(G_{1}T). In the proof of [Jan, II 7.15] adapted for the G_{1}T -translation functor there exists L̂_{1}(η) with L̂_{1}(η) ∈ B̂_{λ}(G_{1}T) such that

T_{λ}^{μ}(L̂_{1}(η)) = L̂_{1}(γ).  (3.4.3)

Hence, by using (3.4.2),

Ext_{G_{1}}^{•}(_{⨁σ∈B_{λ}(G_{1})}L(σ), T_{μ}^{λ}(H^{0}(μ′))) ≅ ⨁_{η∈B_{λ}(G_{1}T)} Ext_{G_{1}T}^{•} (T_{λ}^{μ}(L̂_{1}(η)), H^{0}(μ′))
≅ Ext_{G_{1}T}^{•} (_{⨁η∈B_{λ}(G_{1}T)}T_{λ}^{μ}(L̂_{1}(η)), H^{0}(μ′))
⊇ Ext_{G_{1}T}^{•} (_{⨁γ∈B_{μ}(G_{1}T)}L̂_{1}(γ), H^{0}(μ′))
≅ Ext_{G_{1}}^{•}(_{⨁ζ∈B_{μ}(G_{1})}L_{1}(ζ), H^{0}(μ′))

Therefore, dim V_{G_{1}}(H^{0}(μ′)) ≤ dim V_{G_{1}}(T_{μ}^{λ}(H^{0}(μ′))). In Section 2.4 we have seen that V_{G_{1}}(T_{μ}^{0}(H^{0}(μ′))) ⊆ V_{G_{1}}(H^{0}(μ′)). Since V_{G_{1}}(H^{0}(μ′)) is irreducible, it follows that

V_{G_{1}}(T_{μ}^{0}(H^{0}(μ′))) = V_{G_{1}}(H^{0}(μ′)).
□

3.5 An open question in [CNP, (5.1)] was to determine when there exist indecomposable G-modules M in the principal block for G_{r} (not projective as G_{r}-modules) such that H^{•}(G_{r}, M) = 0. An affirmative answer was given for r ≥ 2 in [BN, (4.1.1) Thm.]. The following theorem provides an affimative answer for r = 1 as long as G does not have underlying root system A_{1}.

(3.5.1) Theorem. Let G be a connected reductive algebraic group with p > h. There exists a G-module M such that (a) M is in the principal block of G_{1};

(b) H^{•}(G_{1}, M) = 0 for • > 0;

(c) V_{G_{1}}(M) = G · 𝔲_{J} where J = {α} ⊆ Δ.

Proof. Let μ′ = w · μ ∈ X(T)_{+} where w ∈ W_{p} and μ ∈ \overline{C}_{ℤ}. Also assume that Φ_{μ} is of type A_{1}. Consider M = T_{μ}^{0}(H^{0}(μ′)). By construction M is in B_{0}(G_{1}). Moreover, by Theorem 3.4.1, V_{G_{1}}(M) = G · 𝔲_{J} where J = {α} ⊆ Δ. We need to show that H^{•}(G_{1}, M) = 0 for • > 0. Now

H^{•}(G_{1}, M) ≅ Ext_{G_{1}}^{•}(k, T_{μ}^{0}(H^{0}(μ′)))
⊆ Ext_{G_{1}}^{•}(k, H^{0}(μ′) ⊗ H^{0}(w(−μ)))
≅ Ext_{G_{1}}^{•}(H^{0}(w(−μ))^{∗}, H^{0}(μ′))
≅ Ext_{G_{1}}^{•}(V (w_{0}wμ), H^{0}(μ′)).

Let L(λ) be a G-composition factor of H^{0}(σ). Then λ ↑ σ and λ ∈ W · σ + pX(T). Decompose λ = λ_{0} +pν and σ = σ_{0} +pν′ where λ_{0}, σ_{0} ∈ X_{1}(T) and ν, ν′ ∈ X(T)_{+}. Observe that L(λ) = L_{1}(λ_{0}) ⊗ L(ν)^{(1)}. Therefore, if L_{1}(λ_{0}) is an G_{1}-composition factor of H^{0}(λ) then λ_{0} ∈ W · σ + pX(T). We can conclude that Ext_{G_{1}}^{•}(V (w_{0}wμ), H^{0}(μ′)) ≠ 0 for some • ≥ 0 implies that w_{0}wμ ∈ W · μ + pX(T).

We can find a fundamental weight ω_{α} for some α ∈ Δ such that −2 ≤ ⟨ω_{α}, β^{∨}⟩ ≤ 2 for all β ∈ Φ. Let μ = −ω_{α}. Then μ ∈ \overline{C}_{ℤ}. Furthermore, Φ_{μ} is of type A_{1}. Also, −μ ∈ X(T)_{+} so w = 1. Suppose that w_{0}wμ = w_{0}μ ∈ W · μ + pX(T). Now

w_{0}μ = w_{0}(−ω_{α}) = −w_{0}(ω_{α}) = ω_{β}

for some β ∈ Δ. We can conclude that ω_{β} = w · (−ω_{α}) + pν for some w ∈ W and ν ∈ X(T). If γ ∈ Δ then

⟨ω_{β}, γ^{∨}⟩ = ⟨w · (−ω_{α}), γ^{∨}⟩ + p⟨ν, γ^{∨}⟩.  (3.5.2)

Hence,

|p⟨ν, γ^{∨}⟩| = |⟨ω_{β}, γ^{∨}⟩ − ⟨w · (−ω_{α}), γ^{∨}⟩|
= |⟨ω_{β}, γ^{∨}⟩ − ⟨−ω_{α} + ρ, w^{−1}γ^{∨}⟩ + 1|
≤ | − ⟨−ω_{α} + ρ, w^{−1}γ^{∨}⟩ + 2|
≤ h.

It follows that for p > h, ν = 0, thus

ω_{β} = w · (−ω_{α}) = w(−ω_{α} + ρ) − ρ.

Therefore, ω_{β} + ρ = w(−ω_{α} + ρ). Since ω_{β} + ρ, −ω_{α} + ρ ∈ X(T)_{+}, we have ω_{β} + ρ = ω_{α} + ρ (i.e. ω_{β} = −ω_{α}). This is a contradiction. Consequently, w_{0}wμ ∉ W · μ + pX(T) and H^{•}(G_{1}, T_{μ}^{0}(H^{0}(μ′))) = 0. _{□}

We can now show that the representation theoretic nucleus for G_{1} for p > h is precisely the subregular orbit.

(3.5.3) Corollary. Let G be a connected reductive algebraic group with p > h. Then θ_{G_{1}} = G · 𝔲_{J} where J = {α} ⊆ Δ.

Proof. From Theorem 3.3.1, we have θ_{G_{1}} ⊆ G · 𝔲_{J}. On the other hand, by Theorem 3.5.1, there exists an M ∈ S with V_{G_{1}}(M) = G · 𝔲_{J}, thus θ_{G_{1}} = G · 𝔲_{J}. _{□}

In view of the equivalence of categories between the G_{1}-modules and the u(𝔤)-modules, Theorem 1.2.1 follows immediately from the above corollary.

4 Further directions

4.1 In this section, we describe some questions opened up by this work. Let 𝔤 be a prestricted Lie algebra.

(4.1.1) Question. What is θ_{𝔤} in the case of the p-restricted Lie algebra of a connected reductive algebraic group with p ≤ h + 1?

(4.1.2) Question. What is θ_{𝔤} for 𝔤 a restricted Lie algebra of Cartan type?

(4.1.3) Question. What is the correct definition of the nucleus Y_{𝔤} in general? It should be defined just in terms of the Lie structure, without mentioning cohomology. There should then be a theorem stating that it is equal to the representation theoretic nucleus θ_{𝔤}.

(4.1.4) Question. Is it true that if M and M′ are finitely generated modules in the principal block of u(𝔤) then V_{𝔤}(M) ∩ V_{𝔤}(M′) ⊆ V_{𝔤}(M, M′) ∪ θ_{𝔤}? This was proved in the finite group case in [C1], but the proof does not obviously translate.

(4.1.5) Question. Describe the thick subcategories of the stable category of finitely generated modules in the principal block of 𝔤. If the situation is analogous to the finite group case [BCRi3], then modulo modules whose variety is contained in the nucleus, one should be able to answer the question in terms of varieties. Each line through the origin in the nucleus should then support an infinite number of thick subcategories.

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Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A.

   E-mail address: djb@byrd.math.uga.edu

Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A.

   E-mail address: nakano@sunfs.math.usu.edu